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23,100 | A wholesaler bought 500 kilograms of a certain type of fruit at a price of 40 yuan per kilogram. According to market forecasts, the selling price $y$ (yuan per kilogram) of this fruit is a function of the storage time $x$ (days), given by $y=60+2x$. However, an average of 10 kilograms of this fruit will be lost each day, and it can be stored for up to 8 days at most. Additionally, the wholesaler needs to spend 40 yuan per day on storage costs.
(1) If the wholesaler sells all the fruit after storing it for 1 day, the selling price of the fruit at that time will be \_\_\_\_\_\_ (yuan per kilogram), and the total profit obtained will be \_\_\_\_\_\_ (yuan);
(2) Let the wholesaler sell all the fruit after storing it for $x$ days, try to find the function relationship between the total profit $w$ (yuan) obtained by the wholesaler and the storage time $x$ (days);
(3) Find the maximum profit that the wholesaler can obtain from operating this batch of fruit. | 11600 | 62.5 |
23,101 | Given the following three statements are true:
I. All freshmen are human.
II. All graduate students are human.
III. Some graduate students are pondering.
Considering the following four statements:
(1) All freshmen are graduate students.
(2) Some humans are pondering.
(3) No freshmen are pondering.
(4) Some of the pondering humans are not graduate students.
Which of the statements (1) to (4) logically follow from I, II, and III? | (2). | 0 |
23,102 | If the ratio of the surface areas of three spheres is 1:4:9, then the ratio of their volumes is ______. | 1 : 8 : 27 | 73.4375 |
23,103 | What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $15!$? | 10 | 57.8125 |
23,104 | The lines containing the altitudes of the scalene triangle \( ABC \) intersect at point \( H \). Let \( I \) be the incenter of triangle \( ABC \), and \( O \) be the circumcenter of triangle \( BHC \). It is known that point \( I \) lies on the segment \( OA \). Find the angle \( BAC \). | 60 | 76.5625 |
23,105 | Given the set \( M = \left\{ x \ | \ 5 - |2x - 3| \in \mathbf{N}^{*} \right\} \), the number of all non-empty proper subsets of \( M \) is? | 510 | 25.78125 |
23,106 | Emily has 8 blue marbles and 7 red marbles. She randomly selects a marble, notes its color, and returns it to the bag. She repeats this process 6 times. What is the probability that she selects exactly three blue marbles? | \frac{3512320}{11390625} | 0.78125 |
23,107 | What is the value of $\frac{(3150-3030)^2}{144}$? | 100 | 29.6875 |
23,108 | Say that an integer $A$ is delicious if there exist several consecutive integers, including $A$, that add up to 2024. What is the smallest delicious integer? | -2023 | 84.375 |
23,109 | Use Horner's Rule to find the value of $v_2$ when the polynomial $f(x) = x^5 + 4x^4 + x^2 + 20x + 16$ is evaluated at $x = -2$. | -4 | 24.21875 |
23,110 | Select 5 people from 3 orthopedic doctors, 4 neurosurgeons, and 5 internists to form an earthquake relief medical team. The number of ways to select such that there is at least one doctor from each specialty is (answer in digits). | 590 | 43.75 |
23,111 | Rectangle $EFGH$ has area $4032.$ An ellipse with area $4032\pi$ passes through points $E$ and $G$ and has its foci at points $F$ and $H$. Determine the perimeter of the rectangle. | 8\sqrt{2016} | 0 |
23,112 | When submitting problems, Steven the troll likes to submit silly names rather than his own. On day $1$ , he gives no
name at all. Every day after that, he alternately adds $2$ words and $4$ words to his name. For example, on day $4$ he
submits an $8\text{-word}$ name. On day $n$ he submits the $44\text{-word name}$ “Steven the AJ Dennis the DJ Menace the Prince of Tennis the Merchant of Venice the Hygienist the Evil Dentist the Major Premise the AJ Lettuce the Novel’s Preface the Core Essence the Young and the Reckless the Many Tenants the Deep, Dark Crevice”. Compute $n$ . | 16 | 25.78125 |
23,113 | Find the smallest composite number that has no prime factors less than 15. | 289 | 74.21875 |
23,114 | The fourth term of a geometric sequence is 512, and the 9th term is 8. Determine the positive, real value for the 6th term. | 128 | 84.375 |
23,115 | Given that $\sqrt {3} \overrightarrow{a}+ \overrightarrow{b}+2 \overrightarrow{c}= \overrightarrow{0}$, and $| \overrightarrow{a}|=| \overrightarrow{b}|=| \overrightarrow{c}|=1$, find the value of $\overrightarrow{a}\cdot ( \overrightarrow{b}+ \overrightarrow{c})$. | -\dfrac{\sqrt{3}}{2} | 30.46875 |
23,116 | Given that \( A \) and \( B \) are two distinct points on the parabola \( y = 3 - x^2 \) that are symmetric with respect to the line \( x + y = 0 \), calculate the distance |AB|. | 3\sqrt{2} | 60.9375 |
23,117 | Given that $\overrightarrow {a}|=4$, $\overrightarrow {e}$ is a unit vector, and the angle between $\overrightarrow {a}$ and $\overrightarrow {e}$ is $\frac {2π}{3}$, find the projection of $\overrightarrow {a}+ \overrightarrow {e}$ on $\overrightarrow {a}- \overrightarrow {e}$. | \frac {5 \sqrt {21}}{7} | 0 |
23,118 | Two circles, one with radius 4 and the other with radius 5, are externally tangent to each other and are circumscribed by a third circle. Calculate the area of the shaded region formed between these three circles. Express your answer in terms of $\pi$. | 40\pi | 37.5 |
23,119 | Compute the sum of the series:
\[ 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2))))) \] | 126 | 72.65625 |
23,120 | Determine all real numbers $q$ for which the equation $x^4 -40x^2 +q = 0$ has four real solutions which form an arithmetic progression | 144 | 93.75 |
23,121 | Compute
\[
\left( 1 + \sin \frac {\pi}{12} \right) \left( 1 + \sin \frac {5\pi}{12} \right) \left( 1 + \sin \frac {7\pi}{12} \right) \left( 1 + \sin \frac {11\pi}{12} \right).
\] | \frac{1}{8} | 0 |
23,122 | Given the polar equation of a circle is $\rho=2\cos \theta$, the distance from the center of the circle to the line $\rho\sin \theta+2\rho\cos \theta=1$ is ______. | \dfrac { \sqrt {5}}{5} | 0 |
23,123 | Given the parabola $C: y^{2}=2px\left(p \lt 0\right)$ passing through the point $A\left(-2,-4\right)$.
$(1)$ Find the equation of the parabola $C$ and its directrix equation.
$(2)$ A line passing through the focus of the parabola, making an angle of $60^{\circ}$ with the $x$-axis, intersects the parabola at points $A$ and $B$. Find the length of segment $AB$. | \frac{32}{3} | 84.375 |
23,124 | A square sheet contains 1000 points, with any three points, including the vertices of the square, not being collinear. Connect some of these points and the vertices of the square with line segments to divide the entire square into smaller triangles (using the connected line segments and square edges as sides, and ensuring that no two segments, except at endpoints, share common points). How many line segments are connected in total? How many triangles are formed in total? | 2002 | 10.15625 |
23,125 | Given the wheel with a circumference of $11$ feet, the speed $r$ in miles per hour for which the time for a complete rotation of the wheel is shortened by $\frac{1}{4}$ of a second is increased by $5$ miles per hour, find the value of $r$. | 10 | 59.375 |
23,126 | Determine the volume of the solid formed by the set of vectors $\mathbf{v}$ such that:
\[\mathbf{v} \cdot \mathbf{v} = \mathbf{v} \cdot \begin{pmatrix} 12 \\ -34 \\ 6 \end{pmatrix}\] | \frac{4}{3} \pi (334)^{3/2} | 2.34375 |
23,127 | If the average of six data points $a_1, a_2, a_3, a_4, a_5, a_6$ is $\bar{x}$, and the variance is 0.20, what is the variance of the seven data points $a_1, a_2, a_3, a_4, a_5, a_6, \bar{x}$? | \frac{6}{35} | 68.75 |
23,128 | Given $f(x) = ax^3 + bx^9 + 2$ has a maximum value of 5 on the interval $(0, +\infty)$, find the minimum value of $f(x)$ on the interval $(-\infty, 0)$. | -1 | 72.65625 |
23,129 | Given the hexagons grow by adding subsequent layers of hexagonal bands of dots, with each new layer having a side length equal to the number of the layer, calculate how many dots are in the hexagon that adds the fifth layer, assuming the first hexagon has only 1 dot. | 61 | 70.3125 |
23,130 | Seven distinct integers are picked at random from the set {1,2,3,...,12}. What is the probability that among those selected, the second smallest number is 4? | \frac{7}{33} | 38.28125 |
23,131 | Given triangle ABC, where a, b, and c are the sides opposite to angles A, B, and C respectively, sin(2C - $\frac {π}{2}$) = $\frac {1}{2}$, and a<sup>2</sup> + b<sup>2</sup> < c<sup>2</sup>.
(1) Find the measure of angle C.
(2) Find the value of $\frac {a + b}{c}$. | \frac {2 \sqrt{3}}{3} | 38.28125 |
23,132 | Circle $\omega_1$ with radius 3 is inscribed in a strip $S$ having border lines $a$ and $b$ . Circle $\omega_2$ within $S$ with radius 2 is tangent externally to circle $\omega_1$ and is also tangent to line $a$ . Circle $\omega_3$ within $S$ is tangent externally to both circles $\omega_1$ and $\omega_2$ , and is also tangent to line $b$ . Compute the radius of circle $\omega_3$ .
| \frac{9}{8} | 2.34375 |
23,133 | The integer solution for the inequality $|2x-m|\leq1$ with respect to $x$ is uniquely $3$ ($m$ is an integer).
(I) Find the value of the integer $m$;
(II) Given that $a, b, c \in R$, if $4a^4+4b^4+4c^4=m$, find the maximum value of $a^2+b^2+c^2$. | \frac{3\sqrt{2}}{2} | 53.90625 |
23,134 | A ball is dropped from a height of 150 feet. Each time it hits the ground, it rebounds to 40% of the height it fell. How many feet will the ball have traveled when it hits the ground the fifth time? | 344.88 | 14.0625 |
23,135 | Given a sequence $\{a_n\}$ where $a_1=1$ and $a_na_{n-1}=a_{n-1}+(-1)^n$ for $n\geqslant 2, n\in\mathbb{N}^*$, find the value of $\frac{a_3}{a_5}$. | \frac{3}{4} | 86.71875 |
23,136 | In a senior high school class, there are two study groups, Group A and Group B, each with 10 students. Group A has 4 female students and 6 male students; Group B has 6 female students and 4 male students. Now, stratified sampling is used to randomly select 2 students from each group for a study situation survey. Calculate:
(1) The probability of exactly one female student being selected from Group A;
(2) The probability of exactly two male students being selected from the 4 students. | \dfrac{31}{75} | 5.46875 |
23,137 | Four of the following test scores are Henry's and the other four are Julia's: 88, 90, 92, 94, 95, 97, 98, 99. Henry's mean score is 94. What is Julia's mean score? | 94.25 | 48.4375 |
23,138 | Let $g(x)$ be a polynomial of degree 2010 with real coefficients, and let its roots be $s_1,$ $s_2,$ $\dots,$ $s_{2010}.$ There are exactly 1010 distinct values among
\[|s_1|, |s_2|, \dots, |s_{2010}|.\] What is the minimum number of real roots that $g(x)$ can have? | 10 | 60.15625 |
23,139 | Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=3$, $|\overrightarrow{b}|=2\sqrt{3}$, and $\overrightarrow{a}\perp(\overrightarrow{a}+\overrightarrow{b})$, find the projection of $\overrightarrow{b}$ in the direction of $\overrightarrow{a}$. | -3 | 19.53125 |
23,140 | How many zeros are there in the last digits of the following number $P = 11\times12\times ...\times 88\times 89$ ? | 18 | 74.21875 |
23,141 | Given the ellipse $\dfrac{{x}^{2}}{16}+ \dfrac{{y}^{2}}{9}=1$, with left and right foci $F_1$ and $F_2$ respectively, and a point $P$ on the ellipse, if $P$, $F_1$, and $F_2$ are the three vertices of a right triangle, calculate the distance from point $P$ to the $x$-axis. | \dfrac{9}{4} | 39.0625 |
23,142 | Select three different numbers from the set {-8, -6, -4, 0, 3, 5, 7} to form a product and determine the smallest possible value. | -280 | 64.84375 |
23,143 | The numbers $1,2,\ldots,9$ are arranged so that the $1$ st term is not $1$ and the $9$ th term is not $9$ . Calculate the probability that the third term is $3$. | \frac{43}{399} | 0 |
23,144 | Given that $\alpha$ is an angle in the second quadrant, and $\cos\left( \frac {\pi}{2}-\alpha\right)= \frac {4}{5}$, then $\tan\alpha= \_\_\_\_\_\_$. | -\frac {4}{3} | 99.21875 |
23,145 | What is the coefficient of $x^5$ in the expansion of $(1 + x + x^2)^9$ ? | 882 | 8.59375 |
23,146 | There is a set of data: $a_{1}=\frac{3}{1×2×3}$, $a_{2}=\frac{5}{2×3×4}$, $a_{3}=\frac{7}{3×4×5}$, $\ldots $, $a_{n}=\frac{2n+1}{n(n+1)(n+2)}$. Let $S_{n}=a_{1}+a_{2}+a_{3}+\ldots +a_{n}$. Find the value of $S_{12}$. To solve this problem, Xiao Ming first simplified $a_{n}$ to $a_{n}=\frac{x}{(n+1)(n+2)}+\frac{y}{n(n+2)}$, and then calculated the value of $S_{12}$ based on the simplified $a_{n}$. Please follow Xiao Ming's approach to first find the values of $x$ and $y$, and then calculate the value of $S_{12}$. | \frac{201}{182} | 7.03125 |
23,147 | Denote by \( f(n) \) the integer obtained by reversing the digits of a positive integer \( n \). Find the greatest integer that is certain to divide \( n^{4} - f(n)^{4} \) regardless of the choice of \( n \). | 99 | 40.625 |
23,148 | David, Ellie, Natasha, and Lucy are tutors in their school science lab. Their working schedule is as follows: David works every fourth school day, Ellie works every fifth school day, Natasha works every sixth school day, and Lucy works every eighth school day. Today, they all happened to be working together. In how many school days from today will they all next tutor together in the lab? | 120 | 88.28125 |
23,149 | If $\dfrac {\cos (\pi-2\alpha)}{\sin (\alpha- \dfrac {\pi}{4})}=- \dfrac { \sqrt {2}}{2}$, then $\sin 2\alpha=$ \_\_\_\_\_\_ . | - \dfrac {3}{4} | 9.375 |
23,150 | Given that EF = 40 units, FG = 30 units, and one diagonal EH = 50 units, calculate the perimeter of parallelogram EFGH. | 140 | 93.75 |
23,151 | If $f(n)$ is the sum of the digits of $n^2+1$ (where $n \in \mathbb{N}^*$). For example, since $14^2+1=197$, and $1+9+7=17$, thus $f(14)=17$. Let $f_1(n)=f(n)$, $f_2(n)=f(f_1(n))$, ..., $f_{k+1}(n)=f(f_k(n))$, where $k \in \mathbb{N}^*$, then $f_{2005}(8)=$ ． | 11 | 64.0625 |
23,152 | Given that the terminal side of angle $θ$ is symmetric to the terminal side of a $480^\circ$ angle with respect to the $x$-axis, and point $P(x,y)$ is on the terminal side of angle $θ$ (not the origin), then the value of $\frac{xy}{{x}^2+{y}^2}$ is equal to __. | \frac{\sqrt{3}}{4} | 75.78125 |
23,153 | In $\triangle ABC$, $AB=1$, $BC=2$, $\angle B=\frac{\pi}{3}$, let $\overrightarrow{AB}=\overrightarrow{a}$, $\overrightarrow{BC}= \overrightarrow{b}$.
(I) Find the value of $(2\overrightarrow{a}-3\overrightarrow{b})\cdot(4\overrightarrow{a}+\overrightarrow{b})$;
(II) Find the value of $|2\overrightarrow{a}-\overrightarrow{b}|$. | 2 \sqrt{3} | 7.8125 |
23,154 | A group of 40 boys and 28 girls stand hand in hand in a circle facing inwards. Exactly 18 of the boys give their right hand to a girl. How many boys give their left hand to a girl? | 18 | 41.40625 |
23,155 | There are 5 balls numbered $(1)$, $(2)$, $(3)$, $(4)$, $(5)$ and 5 boxes numbered $(1)$, $(2)$, $(3)$, $(4)$, $(5)$. Each box contains one ball. The number of ways in which at most two balls have the same number as their respective boxes is $\_\_\_\_\_\_$. | 109 | 60.9375 |
23,156 | A child who does not understand English tries to spell the word "hello" using cards with the letters "e", "o", "h", "l", "l". How many possible incorrect arrangements can there be if the cards cannot be laid horizontally or upside down? | 59 | 51.5625 |
23,157 | Given that the circumferences of the two bases of a cylinder lie on the surface of a sphere $O$ with a volume of $\frac{{32π}}{3}$, the maximum value of the lateral surface area of the cylinder is ______. | 8\pi | 78.90625 |
23,158 | Let $\left\{\left(s_{1}, s_{2}, \cdots, s_{6}\right) \mid s_{i} \in\{0,1\}, i \in \mathbf{N}_{+}, i \leqslant 6\right\}$. For $\forall x, y \in S$, $x=\left(x_{1}, x_{2}, \cdots, x_{6}\right)$ and $y=\left(y_{1}, y_{2}, \cdots, y_{6}\right)$, define:
1. $x=y$ if and only if $\left(x_{1}-y_{1}\right)^{2}+\left(x_{2}-y_{2}\right)^{2}+\cdots+\left(x_{6}-y_{6}\right)^{2}=0$;
2. $x \cdot y = x_{1} y_{1} + x_{2} y_{2} + \cdots + x_{6} y_{6}$.
Given a non-empty set $T \subseteq S$ that satisfies $\forall u, v \in T, u \neq v$, we have $u \cdot v \neq 0$, find the maximum number of elements in the set $T$. | 32 | 36.71875 |
23,159 | In parallelogram \(ABCD\), the angle at vertex \(A\) is \(60^{\circ}\), \(AB = 73\) and \(BC = 88\). The angle bisector of \(\angle ABC\) intersects segment \(AD\) at point \(E\) and ray \(CD\) at point \(F\). Find the length of segment \(EF\).
1. 9
2. 13
3. 12
4. 15 | 15 | 35.15625 |
23,160 | Let \( x_{0} \) be the largest (real) root of the equation \( x^{4} - 16x - 12 = 0 \). Evaluate \( \left\lfloor 10 x_{0} \right\rfloor \). | 27 | 40.625 |
23,161 | What is the value of $\frac13\cdot\frac92\cdot\frac1{27}\cdot\frac{54}{1}\cdot\frac{1}{81}\cdot\frac{162}{1}\cdot\frac{1}{243}\cdot\frac{486}{1}$? | 12 | 4.6875 |
23,162 | The real value of $x$ such that $\frac{81^{x-2}}{9^{x-2}} = 27^{3x+2}$. | -\frac{10}{7} | 94.53125 |
23,163 | Given the ellipse $C$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, passing through point $Q(\sqrt{2}, 1)$ and having the right focus at $F(\sqrt{2}, 0)$,
(I) Find the equation of the ellipse $C$;
(II) Let line $l$: $y = k(x - 1) (k > 0)$ intersect the $x$-axis, $y$-axis, and ellipse $C$ at points $C$, $D$, $M$, and $N$, respectively. If $\overrightarrow{CN} = \overrightarrow{MD}$, find the value of $k$ and calculate the chord length $|MN|$. | \frac{\sqrt{42}}{2} | 10.15625 |
23,164 | Given that in △ABC, the sides opposite to the internal angles A, B, and C are a, b, and c respectively, and $b^{2}=c^{2}+a^{2}- \sqrt {2}ac$.
(I) Find the value of angle B;
(II) If $a= \sqrt {2}$ and $cosA= \frac {4}{5}$, find the area of △ABC. | \frac {7}{6} | 77.34375 |
23,165 | Squares $JKLM$ and $NOPQ$ are congruent, $JM=20$, and $P$ is the midpoint of side $JM$ of square $JKLM$. Calculate the area of the region covered by these two squares in the plane.
A) $500$
B) $600$
C) $700$
D) $800$
E) $900$ | 600 | 21.09375 |
23,166 | Let all possible $2023$ -degree real polynomials: $P(x)=x^{2023}+a_1x^{2022}+a_2x^{2021}+\cdots+a_{2022}x+a_{2023}$ ,
where $P(0)+P(1)=0$ , and the polynomial has 2023 real roots $r_1, r_2,\cdots r_{2023}$ [not necessarily distinct] so that $0\leq r_1,r_2,\cdots r_{2023}\leq1$ . What is the maximum value of $r_1 \cdot r_2 \cdots r_{2023}?$ | 2^{-2023} | 1.5625 |
23,167 | Player A and Player B play a number guessing game. First, Player A thinks of a number denoted as $a$, then Player B guesses the number that Player A is thinking of, and denotes this guessed number as $b$. Both $a$ and $b$ belong to the set $\{1,2,3,4,5,6\}$. If $|a-b| \leqslant 1$, it is said that "Player A and Player B are in sync". Now, find the probability that two randomly chosen players are "in sync" in this game. | \frac{4}{9} | 96.09375 |
23,168 | Patrícia wrote, in ascending order, the positive integers formed only by odd digits: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 31, 33, ... What was the 157th number she wrote? | 1113 | 1.5625 |
23,169 | Given the function $f(x)=\sin(\omega x+\varphi)$ is monotonically increasing on the interval ($\frac{π}{6}$,$\frac{{2π}}{3}$), and the lines $x=\frac{π}{6}$ and $x=\frac{{2π}}{3}$ are the two symmetric axes of the graph of the function $y=f(x)$, evaluate $f(-\frac{{5π}}{{12}})$. | \frac{\sqrt{3}}{2} | 14.84375 |
23,170 | Given an ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity of $\frac{1}{2}$, a circle with the origin as its center and the short axis of the ellipse as its radius is tangent to the line $\sqrt{7}x-\sqrt{5}y+12=0$.
(1) Find the equation of the ellipse $C$;
(2) Let $A(-4,0)$, and a line $l$ passing through point $R(3,0)$ and intersecting with the ellipse $C$ at points $P$ and $Q$. Connect $AP$ and $AQ$ intersecting with the line $x=\frac{16}{3}$ at points $M$ and $N$, respectively. If the slopes of lines $MR$ and $NR$ are $k_{1}$ and $k_{2}$, respectively, determine whether $k_{1}k_{2}$ is a constant value. If it is, find this value; otherwise, explain the reason. | -\frac{12}{7} | 42.96875 |
23,171 | Given an arithmetic sequence $\{a_{n}\}$ with the sum of the first $n$ terms as $S_{n}$, and a positive geometric sequence $\{b_{n}\}$ with the sum of the first $n$ terms as $T_{n}$, where $a_{1}=2$, $b_{1}=1$, and $b_{3}=3+a_{2}$. <br/>$(1)$ If $b_{2}=-2a_{4}$, find the general formula for the sequence $\{b_{n}\}$; <br/>$(2)$ If $T_{3}=13$, find $S_{3}$. | 18 | 25 |
23,172 | Four students are admitted to three universities. Find the probability that each university admits at least one student. | \frac{4}{9} | 89.84375 |
23,173 | Given in the polar coordinate system, point P moves on the curve $\rho^2\cos\theta-2\rho=0$, the minimum distance from point P to point $Q(1, \frac{\pi}{3})$ is \_\_\_\_\_\_. | \frac{3}{2} | 67.96875 |
23,174 | Let $f(x)$ be the function such that $f(x)>0$ at $x\geq 0$ and $\{f(x)\}^{2006}=\int_{0}^{x}f(t) dt+1.$
Find the value of $\{f(2006)\}^{2005}.$ | 2006 | 71.875 |
23,175 | Find the ratio of the area of $\triangle BCY$ to the area of $\triangle ABY$ in the diagram if $CY$ bisects $\angle BCA$. [asy]
import markers;
real t=34/(34+28);
pair A=(-17.18,0);
pair B=(13.82,0);
pair C=(0,30);
pair Y=t*B+(1-t)*A;
draw(C--A--B--C--Y);
label("$A$",A,SW);
label("$B$",B,E);
label("$C$",C,N);
label("$Y$",Y,NE);
label("$32$",.5*(B+A),S);
label("$34$",.5*(B+C),NE);
label("$28$",.5*(A+C),NW);
[/asy] | \frac{17}{14} | 38.28125 |
23,176 | The sum of an infinite geometric series is $64$ times the series that results if the first four terms of the original series are removed. What is the value of the series' common ratio? | \frac{1}{2} | 39.0625 |
23,177 | Given the numbers $1$, $2$, $3$, $4$, $5$, randomly select $3$ numbers (with repetition allowed) to form a three-digit number, find the probability that the sum of its digits equals $12$. | \dfrac{2}{25} | 6.25 |
23,178 | In the Cartesian coordinate system xOy, point P(x0, y0) is on the unit circle O. Suppose the angle ∠xOP = α, and if α ∈ (π/3, 5π/6), and sin(α + π/6) = 3/5, determine the value of x0. | \frac{3-4\sqrt{3}}{10} | 29.6875 |
23,179 | Schools A and B are having a sports competition with three events. In each event, the winner gets 10 points and the loser gets 0 points, with no draws. The school with the highest total score after the three events wins the championship. It is known that the probabilities of school A winning the three events are 0.5, 0.4, and 0.8, respectively, and the results of each event are independent.<br/>$(1)$ Find the probability of school A winning the championship;<br/>$(2)$ Let $X$ represent the total score of school B, find the probability distribution and expectation of $X$. | 13 | 21.875 |
23,180 | A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is planning to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered. | 12\% | 69.53125 |
23,181 | Given the function $$f(x)=a+\sin(x-\frac{1}{2})$$, if $$f(0)+f(\frac{1}{2019})+f(\frac{2}{2019})+…+f(\frac{2017}{2019})+f(\frac{2018}{2019})+f(1)=1010$$, find the value of the real number $a$. | \frac{1}{2} | 94.53125 |
23,182 | Inside of the square $ABCD$ the point $P$ is given such that $|PA|:|PB|:|PC|=1:2:3$ . Find $\angle APB$ . | 135 | 28.125 |
23,183 | Given that a water tower stands 60 meters high and contains 150,000 liters of water, and a model of the tower holds 0.15 liters, determine the height of Liam's model tower. | 0.6 | 64.84375 |
23,184 | The sum of three numbers \(a\), \(b\), and \(c\) is 150. If we increase \(a\) by 10, decrease \(b\) by 3, and multiply \(c\) by 4, the three resulting numbers are equal. What is the value of \(b\)? | \frac{655}{9} | 80.46875 |
23,185 | A rectangular prism has 6 faces, 12 edges, and 8 vertices. If a new pyramid is added using one of its rectangular faces as the base, calculate the maximum value of the sum of the exterior faces, vertices, and edges of the resulting shape after the fusion of the prism and pyramid. | 34 | 23.4375 |
23,186 | Given that $F_{1}$ and $F_{2}$ are two foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{7}=1$, $A$ is a point on the ellipse, and $\angle AF_{1}F_{2}=45^{\circ}$, calculate the area of triangle $AF_{1}F_{2}$. | \frac{7}{2} | 24.21875 |
23,187 | A square and a regular pentagon have the same perimeter. Let $C$ be the area of the circle circumscribed about the square, and $D$ the area of the circle circumscribed around the pentagon. Find $C/D$.
A) $\frac{25}{128}$
B) $\frac{25(5 + 2\sqrt{5})}{128}$
C) $\frac{25(5-2\sqrt{5})}{128}$
D) $\frac{5\sqrt{5}}{128}$ | \frac{25(5-2\sqrt{5})}{128} | 41.40625 |
23,188 | The repeating decimal for $\frac{5}{13}$ is $0.cdc\ldots$ What is the value of the sum $c+d$? | 11 | 91.40625 |
23,189 | Given the function $f(x)=\sin (2x+φ)$, where $|φ| < \dfrac{π}{2}$, the graph is shifted to the left by $\dfrac{π}{6}$ units and is symmetric about the origin. Determine the minimum value of the function $f(x)$ on the interval $[0, \dfrac{π}{2}]$. | -\dfrac{ \sqrt{3}}{2} | 74.21875 |
23,190 | Calculate the product of $\frac{5}{3} \times \frac{6}{5} \times \frac{7}{6} \times \cdots \times \frac{2010}{2009}$. | 670 | 96.09375 |
23,191 | Given that $a>0$, $b>1$, and $a+b=2$, find the minimum value of $$\frac{1}{2a}+\frac{2}{b-1}$$. | \frac{9}{2} | 54.6875 |
23,192 | What is the area enclosed by the graph of $|x| + |3y| = 9$? | 54 | 85.9375 |
23,193 | Given the function $f(x)= \dfrac {x+3}{x+1}$, let $f(1)+f(2)+f(4)+f(8)+f(16)=m$ and $f( \dfrac {1}{2})+f( \dfrac {1}{4})+f( \dfrac {1}{8})+f( \dfrac {1}{16})=n$, then $m+n=$ \_\_\_\_\_\_. | 18 | 26.5625 |
23,194 | In triangle $XYZ$, $XY=12$, $YZ=16$, and $XZ=20$. Point $M$ is on $\overline{XY}$, $N$ is on $\overline{YZ}$, and $O$ is on $\overline{XZ}$. Let $XM = p \cdot XY$, $YN = q \cdot YZ$, and $ZO = r \cdot XZ$, where $p$, $q$, and $r$ are positive and satisfy $p+q+r=3/4$ and $p^2+q^2+r^2=1/2$. The ratio of the area of triangle $MNO$ to the area of triangle $XYZ$ can be written as $s/t$, where $s$ and $t$ are relatively prime positive integers. Find $s+t$. | 41 | 29.6875 |
23,195 | Given a complex number $z=3+bi$ ($b\in\mathbb{R}$), and $(1+3i) \cdot z$ is a pure imaginary number.
(1) Find the complex number $z$;
(2) If $w= \frac{z}{2+i}$, find the modulus of the complex number $w$. | \sqrt{2} | 97.65625 |
23,196 | Given positive integers \(a\) and \(b\) such that \(15a + 16b\) and \(16a - 15b\) are both perfect squares, find the smallest possible value of these two perfect squares. | 231361 | 3.125 |
23,197 | During the military training for new freshman students, after two days of shooting practice, student A can hit the target 9 times out of 10 shots, and student B can hit the target 8 times out of 9 shots. A and B each take a shot at the same target (their shooting attempts do not affect each other). Determine the probability that the target is hit. | \frac{89}{90} | 87.5 |
23,198 | A deck of 100 cards is numbered from 1 to 100, each card having the same number printed on both sides. One side of each card is red and the other side is yellow. Barsby places all the cards, red side up, on a table. He first turns over every card that has a number divisible by 2. He then examines all the cards, and turns over every card that has a number divisible by 3. Determine the number of cards that have the red side up when Barsby is finished. | 49 | 40.625 |
23,199 | For the system of equations \(x^{2} + x^{2} y^{2} + x^{2} y^{4} = 525\) and \(x + x y + x y^{2} = 35\), find the sum of the real y values that satisfy the equations. | \frac{5}{2} | 3.125 |
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